Thursday, September 25, 2014
Tuesday, September 23, 2014
Physical Systems can be divided up into a number of different catagories, depending on particular properties that the system exhibits. Some of these system classifications are very easy to work with, and have a large theory base for studying. Some system classifications are very complex, and have still not been investigated with any degree of success. This book will focus primarily on linear time-invariant (LTI) systems. LTI systems are the easiest class of system to work with, and have a number of properties that make them ideal to study.
Systems
We will begin our study by talking about systems. Systems, in the barest sense, are devices that take input, and produce an output. The output is related to the input by a certain relation known as the system response. The system response usually can be modeled with a mathematical relationship between the system input and the system output.
There are many different types of systems, and the process of classifying systems in these ways is called system identification.
Friday, September 19, 2014
MATLAB is a programming tool that is commonly used in the field of control engineering. We will not consider MATLAB in the main narrative of this book, but we will provide an appendix that will show how MATLAB is used to solve control problems, and design and model control systems. This appendix can be found at: Control Systems/MATLAB. For more information on MATLAB in general, see: MATLAB Programming Nearly all textbooks on the subject of control systems, linear systems, and system analysis will use MATLAB as an integral part of the text. Students who are learning this subject at an accredited university will certainly have seen this material in their textbooks, and are likely to have had MATLAB work as part of their classes. It is from this perspective that the MATLAB appendix is written. There are a number of other software tools that are useful in the analysis and design of control systems. Additional information can be added in the appendix of this book, depending on the experiance and prior knowledge of contributors.
Here we are going to give a brief listing of the various different methodologies within the sphere of control
engineering. Oftentimes, the lines between these methodologies are blurred, or even erased completely.
Classical Controls
Control methodologies where the ODEs that describe a system are transformed using the Laplace, Fourier,
or Z Transforms, and manipulated in the transform domain.
Modern Controls
Methods where high-order differential equations are broken into a system of first-order equations. The
input, output, and internal states of the system are described by vectors called "state variables".
Robust Control
Control methodologies where arbitrary outside noise/disturbances are accounted for, as well as internal
inaccuracies caused by the heat of the system itself, and the environment.
Optimal Control
In a system, performance metrics are identified, and arranged into a "cost function". The cost function is
minimized to create an operational system with the lowest cost.
Adaptive Control
In adaptive control, the control changes it's response characteristics over time to better control the system.
Nonlinear Control
The youngest branch of control engineering, nonlinear control encompasses systems that cannot be
described by linear equations or ODEs, and for which there is often very little supporting theory available.
Game Theory
Game Theory is a close relative of control theory, and especially robust control and optimal control
theories. In game theory, the external disturbances are not considered to be random noise processes, but
instead are considered to be "opponents". Each player has a cost function that they attempt to minimize,
and that their opponents attempt to maximize.
engineering. Oftentimes, the lines between these methodologies are blurred, or even erased completely.
Classical Controls
Control methodologies where the ODEs that describe a system are transformed using the Laplace, Fourier,
or Z Transforms, and manipulated in the transform domain.
Modern Controls
Methods where high-order differential equations are broken into a system of first-order equations. The
input, output, and internal states of the system are described by vectors called "state variables".
Robust Control
Control methodologies where arbitrary outside noise/disturbances are accounted for, as well as internal
inaccuracies caused by the heat of the system itself, and the environment.
Optimal Control
In a system, performance metrics are identified, and arranged into a "cost function". The cost function is
minimized to create an operational system with the lowest cost.
Adaptive Control
In adaptive control, the control changes it's response characteristics over time to better control the system.
Nonlinear Control
The youngest branch of control engineering, nonlinear control encompasses systems that cannot be
described by linear equations or ODEs, and for which there is often very little supporting theory available.
Game Theory
Game Theory is a close relative of control theory, and especially robust control and optimal control
theories. In game theory, the external disturbances are not considered to be random noise processes, but
instead are considered to be "opponents". Each player has a cost function that they attempt to minimize,
and that their opponents attempt to maximize.
The field of control systems started essentially in the ancient world. Early civilizations, notably the greeks and the arabs were heaviliy preoccupied with the accurate measurement of time, the result of which were several "water clocks" that were designed and implemented. However, there was very little in the way of actual progress made in the field of engineering until the beginning of the renassiance in Europe. Leonhard Euler (for whom Euler's Formula is named) discovered a powerful integral transform, but Pierre Simon-Laplace used the transform (later called the Laplace Transform) to solve complex problems in probability theory. Joseph Fourier was a court mathematician in France under Napoleon I. He created a special function decomposition called the Fourier Series, that was later generalized into an integral transform, and named in his honor (the Fourier Transform).
The "golden age" of control engineering occured between 1910-1945, where mass communication methods were being created and two world wars were being fought. During this period, some of the most famous names in controls engineering were doing their work: Nyquist and Bode. Hendrik Wade Bode and Harry Nyquist, especially in the 1930's while working with Bell Laboratories, created the bulk of what we now call "Classical Control Methods". These methods were based off the results of the Laplace and Fourier Transforms, which had been previously known, but were made popular by Oliver Heaviside around the turn of the century. Previous to Heaviside, the transforms were not widely used, nor respected mathematical tools. Bode is credited with the "discovery" of the closed-loop feedback system, and the logarithmic plotting technique that still bears his name (bode plots). Harry Nyquist did extensive research in the field of system stability and information theory. He created a powerful stability criteria that has been named for him (The Nyquist Criteria).
Modern control methods were introduced in the early 1950's, as a way to bypass some of the shortcomings of the classical methods. Modern control methods became increasingly popular after 1957 with the invention of the computer, and the start of the space program. Computers created the need for digital control methodologies, and the space program required the creation of some "advanced" control techniques, such as "optimal control", "robust control", and "nonlinear control". These last subjects, and several more, are still active areas of study among research engineers.
Modern control methods were introduced in the early 1950's, as a way to bypass some of the shortcomings of the classical methods. Modern control methods became increasingly popular after 1957 with the invention of the computer, and the start of the space program. Computers created the need for digital control methodologies, and the space program required the creation of some "advanced" control techniques, such as "optimal control", "robust control", and "nonlinear control". These last subjects, and several more, are still active areas of study among research engineers.
1749-1827
1768-1840
Oliver Heaviside
The study and design of automatic Control Systems, a field known as control engineering, is a large and expansive area of study. Control systems, and control engineering techniques have become a pervasive part of modern technical society. From devices as simple as a toaster, to complex machines like space shuttles and rockets, control engineering is a part of our everyday life. This book will introduce the field of control engineering, and will build upon those foundations to explore some of the more advanced topics in the field. Note, however, that control engineering is a very large field, and it would be foolhardy of any author to think that they could include all the information into a single book. Therefore, we will be content here to provide the foundations of control engineering, and then describe some of the more advanced topics in the field.
Control systems are components that are added to other components, to increase functionality, or to meet a set of design criteria.
Control System
A Control System is a device, or a collection of devices that manage the behavior of other devices. Some devices are not controllable. A control system is an interconnection of components connected or related in such a manner as to command, direct, or regulate itself or another system.
Controller
A controller is a control system that manages the behavior of another device or system.
Compensator
A Compensator is a control system that regulates another system, usually by conditioning the input or the output to that system. Compensators are typically employed to correct a single design flaw, with the intention of affecting other aspects of the design in a minimal manner.
Classical and Modern
Classical and Modern control methodologies are named in a misleading way, because the group of techniques called "Classical" were actually developed later then the techniques labled "Modern". However, in terms of developing control systems, Modern methods have been used to great effect more recently, while the Classical methods have been gradually falling out of favor. Most recently, it has been shown that Classical and Modern methods can be combined to highlight their respective strengths and weaknesses.
Classical Methods, which this book will consider first, are methods involving the Laplace Transform domain. Physical systems are modeled in the so-called "time domain", where the response of a given system is a function of the various inputs, the previous system values, and time. As time progresses, the state of the system, and it's response change. However, time-domain models for systems are frequently modeled using high-order differential equations, which can become impossibly difficult for humans to solve, and some of which can even become impossible for modern computer systems to solve efficiently. To counteract this problem, integral transforms, such as the Laplace Transform, and the Fourier Transform can be employed to change an Ordinary Differential Equation (ODE) in the time domain into a regular algebraic polynomial in the transform domain. Once a given system has been converted into the transform domain, it can be manipulated with greater ease, and analyzed quickly and simply, by humans and computers alike.
Modern Control Methods, instead of changing domains to avoid the complexities of time-domain ODE mathematics, converts the differential equations into a system of lower-order time domain equations called State Equations, which can then be manipulated using techniques from linear algebra (matrices). This book will consider Modern Methods second.
A third distinction that is frequently made in the realm of control systems is to divide analog methods (classical and modern, described above) from digital methods. Digital Control Methods were designed to try and incorporate the emerging power of computer systems into previous control methodologies. A special transform, known as the Z-Transform, was developed that can adequately describe digital systems, but at the same time can be converted (with some effort) into the Laplace domain. Once in the Laplace domain, the digital system can be manipulated and analyzed in a very similar manner to Classical analog systems. For this reason, this book will not make a hard and fast distinction between Analog and Digital systems, and instead will attempt to study both paradigms in parallel.
Classical and Modern
Classical and Modern control methodologies are named in a misleading way, because the group of techniques called "Classical" were actually developed later then the techniques labled "Modern". However, in terms of developing control systems, Modern methods have been used to great effect more recently, while the Classical methods have been gradually falling out of favor. Most recently, it has been shown that Classical and Modern methods can be combined to highlight their respective strengths and weaknesses.
Classical Methods, which this book will consider first, are methods involving the Laplace Transform domain. Physical systems are modeled in the so-called "time domain", where the response of a given system is a function of the various inputs, the previous system values, and time. As time progresses, the state of the system, and it's response change. However, time-domain models for systems are frequently modeled using high-order differential equations, which can become impossibly difficult for humans to solve, and some of which can even become impossible for modern computer systems to solve efficiently. To counteract this problem, integral transforms, such as the Laplace Transform, and the Fourier Transform can be employed to change an Ordinary Differential Equation (ODE) in the time domain into a regular algebraic polynomial in the transform domain. Once a given system has been converted into the transform domain, it can be manipulated with greater ease, and analyzed quickly and simply, by humans and computers alike.
Modern Control Methods, instead of changing domains to avoid the complexities of time-domain ODE mathematics, converts the differential equations into a system of lower-order time domain equations called State Equations, which can then be manipulated using techniques from linear algebra (matrices). This book will consider Modern Methods second.
A third distinction that is frequently made in the realm of control systems is to divide analog methods (classical and modern, described above) from digital methods. Digital Control Methods were designed to try and incorporate the emerging power of computer systems into previous control methodologies. A special transform, known as the Z-Transform, was developed that can adequately describe digital systems, but at the same time can be converted (with some effort) into the Laplace domain. Once in the Laplace domain, the digital system can be manipulated and analyzed in a very similar manner to Classical analog systems. For this reason, this book will not make a hard and fast distinction between Analog and Digital systems, and instead will attempt to study both paradigms in parallel.
Subscribe to:
Posts (Atom)
